I'd disagree, lp/mm is not useful as a small sensor will win over a big one (it has smaller pixels to resolve detail and a smaller lens where it's easier to get more resolution per mm, essential even).

What you see in the final picture is the whole image. Let's say I have a phone which manages more lp/mm than a Canon 1Dx. Let's shoot a pic of someone's face which is 1/2 the picture height in both cases (same framing). Let's hypothesize that the phone gets 80 lp/mm over its 3mm high sensor and the 1Dx with a £2000 L series lens 50 lp/mm over its 24mm one. On your argument the phone is better (80 vs. 50).

However that means the Canon has 600 lp of resolution over the face's height and the phone 120 lp. I'm thinking the Canon will look a lot sharper.

However if using lp/ph you will see the Canon (1200) is sharper than the phone (240) and that will agree with what you see in the pictures, in direct proportion.